All of the 5th grade teachers and students from Springer went on a field trip to an art museum. Tickets were $$5.50$ each for teachers and $$3.50$ each for students, and the group paid $$35.50$ in total. A few weeks later, the same group visited a science museum where the tickets cost $$22.00$ each for teachers and $$9.00$ each for students, and the group paid $$107.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+3.5y = 35.5}$ ${22x+9y = 107}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-22x-14y = -142}$ ${22x+9y = 107}$ Add the top and bottom equations together. $ -5y = -35 $ $ y = \dfrac{-35}{-5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {5.5x+3.5y = 35.5}$ to find $x$ ${5.5x + 3.5}{(7)}{= 35.5}$ $5.5x+24.5 = 35.5$ $5.5x = 11$ $x = \dfrac{11}{5.5}$ ${x = 2}$ You can also plug ${y = 7}$ into $ {22x+9y = 107}$ and get the same answer for $x$ ${22x + 9}{(7)}{= 107}$ ${x = 2}$ There were $2$ teachers and $7$ students on the field trips.